I want the smallest possible hung-window frame that can hold my window unit (air conditioner).
Window frame size would be 20.5″ W x 16″ H.

Normally the bottom rail of the sash holds in the air conditioner’s top guide.
Instead of a complete sash, only the bottom rail would fit above the air conditioner.
(I am using terms from https://www.salisburyjoinery.com/blog/parts-of-a-window-explained > bottom rail, stile, jamb)

There are two ways this bottom rail could be fabricated:
– Fabricate a bottom rail from wood.
– Cut a length of stile to fit across the window into the left & right jambs.
    Would a length of stile be strong enough to hold in the air conditioner?
Has anyone installed an air conditioner like that?

How do I find a window-frame manufacturer that would build such an odd window frame?
How do I go about locating a window installer?
Would the window installer know where to get a custom frame like that?

The window frame will go into an existing blank wall with 2×4 studs, 1/2″ drywall, and stucco siding.

I am not installing a through-the-wall air conditioner because:
– the smallest through-the-wall unit is 8,000 BTU
– I already have a 6,000 BTU window unit
– this small room needs only 3,000 BTU
I ran the 6,000 BTU window unit in the room’s only window last summer.
On the hottest afternoon, it cycled off half the time.
So I figure 3,000 BTU would be enough.